Recent and Current Undergraduate Research Projects

• Coupled Reaction-Diffusion Equations with Nonlinear Diffusion
• Exploration of Phase and Defect-mediated Turbulence in the Complex Ginzburg-Landau Equation
• Simulation of a Low-Reynolds Number Swimmer
• Nonlinear Time Series Analysis

Coupled Reaction-Diffusion Equations with Nonlinear Diffusion

This project consists in a numerical simulation of reaction-diffusion equations with nonlinear diffusion. Many population dynamics models involve such equations, in which at least two dependent variables are to be considered, one for the density of nutrients and one for the species that consumes these nutrients. The project will be based on a model proposed by Kawasaki and co-workers [1], for which existence results for traveling wave solutions are known [2]. A code will be made available to the undergraduate student. Depending on his/her qualification, the student will then have the possibility to learn the numerical method used in this code, and/or numerically explore the parameter space of the model. The goal will be to obtain results on the stability of traveling wave solutions, and to quantify the effect of different types of nonlinear diffusion.

Exploration of Phase and Defect-mediated Turbulence in the Complex Ginzburg-Landau Equation

The fact that phase turbulence [3, 4] in the complex Ginzburg-Landau equation leads to defect-mediated turbulence [5, 6] is fairly well established [7, 8]. What is less clear however is whether such a transition can occur arbitrarily close to the phase turbulence threshold. Numerical explorations seem to indicate that the answer depends on the dimension of the system: in one spatial dimension, there seems to be a region in the parameter space where the system undergoes phase turbulence but not defect-mediated turbulence, regardless of the size of the system [9]. In two dimensions, it is believed that such a region does not exist and that phase turbulence always leads to defect-mediated turbulence in finite (but possibly large) time and for systems of large enough size [10]. The proposed undergraduate project consists in exploring these ideas and getting an intuition for the dynamics of the complex Ginzburg-Landau equation.

Student: John Pate

Simulation of a Low-Reynolds Number Swimmer

This project consists in (i) understanding issues related to "life at low Reynolds numbers" [11], (ii) reading and understanding a recent paper describing a simple low-Reynolds number swimmer consisting of three linked spheres [12], (iii) understanding the methods of regularized Stokeslets [13], and (iv) simulating the motion of the above-mentioned low-Reynolds-number swimmer using this method. The project will introduce the student to low-Reynolds-number hydrodynamics and to the numerical solving of linear partial differential equations. The method of regularized Stokeslets [13] relies on the principle of superposition to solve the Stokes equation by means of a distribution of smoothed-out point forces acting on immersed boundaries. The strength of these forces can either be imposed, or calculated so that boundary conditions are satisfied. The goal of the project is to simulate the swimmer's motion and tabulate its speed as a function of its geometry.

Student: Dan Norwood (now working for Google)

Nonlinear Time Series Analysis

This project consists in learning methods for the analysis of nonlinear time series, and to explore applications of these techniques to biology. The student will first familiarize himself/herself with two papers describing the application of nonlinear time series analysis to the human electrocardiogram [14] and to the human gait [15]. He/she will then develop codes to reproduce these analyses and then do original work on data sets obtained from researchers in the Department of Molecular and Cellular Biology.

Student: Harish Anandhanarayanan
Report (PDF)
Online presentation (PDF)

References

1. K. Kawasaki, A. Mochizuchi, M. Matsushita, T. Umeda and N. Shigesada, Modeling spatio-temporal patterns generated by Bacillus subtilis, J. Theor. Biol. 188, 177-185 (1997).
2. R.A. Satnoianu, P.K. Maini, F.S. Garduno and J.P. Armitage, Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation, Discrete and Continuous Dynamical Systems B 1, 339-362 (2001).
3. Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys. 55, 356-369 (1976).
4. Y. Pomeau and P. Manneville, Stability and fluctuations of a spatially periodic convective flow, J. Physique (Paris) Lettres 40, L609-L612 (1979).
5. P. Coullet and J. Lega, Defect-mediated turbulence in wave patterns, Europhys. Lett. 7, 511-516 (1988).
6. P. Coullet, L. Gil, and J. Lega, Defect-mediated turbulence, Phys. Rev. Lett. 62, 1619-1622 (1989).
7. H. Chaté and P. Manneville, Phase diagram of the two-dimensional complex Ginzburg-Landau equation, Physica A 224, 348-368 (1996).
8. J. Lega, Traveling hole solutions of the complex Ginzburg-Landau equation: a review, Physica D 152-153, 269-287 (2001).
9. B.I. Schraiman, A. Pumir, W. van Saarloos, P.C. Hohenberg, H. Chaté, and M. Holen, Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation, Physica D 57, 241-248 (1992).
10. P. Manneville and H. Chaté, Phase turbulence in the two-dimensional complex Ginzburg-Landau equation, Physica D 96, 30-46 (1996).
11. E.M. Purcell, Life at low Reynolds number, Am. J. Phys. 45, 3-11 (1977).
12. A. Najafi and R. Golestanian, Simple swimmer at low Reynolds number: Three linked spheres, Phys. Rev. E 69, 062901 (2004).
13. R. Cortez, The method of regularized Stokeslets, SIAM J. Comput. 23, 1204-1225 (2001).
14. M. Perc, Nonlinear time series analysis of the human electrocardiogram, Eur. J. Phys. 26, 757-768 (2005).
15. M. Perc, The dynamics of human gait, Eur. J. Phys. 26, 525-534 (2005).